Capillary surfaces arising in singular perturbation problems. (English) Zbl 1435.35030

Summary: We prove some Bernstein-type theorems for a class of stationary points of the Alt-Caffarelli functional in \(\mathbb{R}^2\) and \(\mathbb{R}^3\) arising as limits of the singular perturbation problem \[\begin{cases} \triangle u_\varepsilon(x)=\beta_\varepsilon(u_\varepsilon)& \text{in } B_1, \\ |u_\varepsilon|\le 1&\text{in } B_1, \ \end{cases}\] in the unit ball \(B_1\) as \(\varepsilon\to 0\). Here \(\beta_\varepsilon(t)=(1/\varepsilon)\beta(t/\varepsilon)\ge 0\), \( \beta\in C_0^\infty[0,1]\), \( \int_0^1\beta(t)\,dt=M> 0\), is an approximation of the Dirac measure and \(\varepsilon>0\). The limit functions \(u=\lim_{\varepsilon_j \to 0}u_{\varepsilon_j}\) of uniformly converging sequences \(\{u_{\varepsilon_j}\}\) solve a Bernoulli-type free boundary problem in some weak sense. Our approach has two novelties: First we develop a hybrid method for stratification of the free boundary \(\partial\{u_0> 0\}\) of blow-up solutions which combines some ideas and techniques of viscosity and variational theory. An important tool we use is a new monotonicity formula for the solutions \(u_\varepsilon\) based on a computation of J. Spruck. It implies that any blow-up \(u_0\) of \(u\) either vanishes identically or is a homogeneous function of degree 1, that is, \(u_0=rg(\sigma)\), \( \sigma\in \mathbb{S}^{N-1}\), in spherical coordinates \((r, \theta)\). In particular, this implies that in two dimensions the singular set is empty at the nondegenerate points, and in three dimensions the singular set of \(u_0\) is at most a singleton. Second, we show that the spherical part \(g\) is the support function (in Minkowski’s sense) of some capillary surface contained in the sphere of radius \(\sqrt{2M}\). In particular, we show that \(\nabla u_0:\mathbb{S}^2\to \mathbb{R}^3\) is an almost conformal and minimal immersion and the singular Alt-Caffarelli example corresponds to a piece of catenoid which is a unique ring-type stationary minimal surface determined by the support function \(g\).


35B25 Singular perturbations in context of PDEs
35J61 Semilinear elliptic equations
49Q05 Minimal surfaces and optimization
35R35 Free boundary problems for PDEs
Full Text: DOI arXiv


[1] ; Alexandroff, Rec. Math. N.S. [Mat. Sbornik], 6, 167 (1939)
[2] 10.1515/crll.1981.325.105
[3] ; Alt, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4), 11, 1 (1984)
[4] 10.2307/1999245
[5] 10.4171/RMI/47 · Zbl 0676.35085
[6] 10.1002/cpa.3160420105 · Zbl 0676.35086
[7] ; Caffarelli, Differential Integral Equations, 8, 1585 (1995)
[8] 10.1090/gsm/068
[9] 10.1512/iumj.1997.46.1470 · Zbl 0909.35012
[10] 10.2307/121117 · Zbl 0960.35112
[11] 10.1090/conm/350/06339 · Zbl 1330.35545
[12] ; Fang, Lectures on minimal surfaces in ℝ3. Proc. Centre Math. Appl. Austr. Nat. Univ., 35 (1996) · Zbl 0955.53002
[13] 10.2140/pjm.2011.250.319 · Zbl 1211.35207
[14] 10.1007/s12220-017-9862-8 · Zbl 1391.35429
[15] 10.1007/s00039-015-0335-6 · Zbl 1326.49078
[16] 10.2140/pjm.2013.265.85 · Zbl 1279.31003
[17] 10.1215/S0012-7094-88-05736-5 · Zbl 0667.49024
[18] ; Nitsche, J. Math. Mech., 11, 293 (1962)
[19] 10.1007/BF00281743 · Zbl 0572.52005
[20] 10.5565/PUBLMAT_36192_03 · Zbl 0773.53004
[21] 10.1007/BF01263611 · Zbl 0912.53009
[22] 10.1017/CBO9781139003858 · Zbl 1287.52001
[23] 10.4310/jdg/1214438183 · Zbl 0575.53037
[24] 10.1007/BF00250468 · Zbl 0222.31007
[25] 10.1080/03605308308820317 · Zbl 0534.35055
[26] 10.1007/s00039-014-0268-5 · Zbl 1295.35344
[27] 10.1007/s00526-002-0171-z
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.